Clare Murphy-Putnam
Chapter 5
5.1 Relationships in Triangles
1) Perpendicular Bisector: a line, segment or ray that bisects a side of a triangle
at a perpendicular angel.
a. Any point on the perpendicular bisector of a segment is equidistance
from the endpoint of the segment
b. Any point equidistant from the endpoints of a segment lies on the
perpendicular bisector of the segment
2) When three or more lines intersect at a common point, the lines are called
concurrent lines, and their point of intersection is called the point of
concurrence. The point of concurrency of the perpendicular bisector of a
triangle is called the circumcenter
3) There are three angle bisectors in any triangle. The angle bisectors of a
triangle are concurrent, and their point of intersection is called the incenter
of the triangle.
4) A median is a segment whose endpoints are a vertex of a triangle and the
midpoints of the side opposite the vertex. Every triangle has three medians.
a. The point of concurrency for the medians of a triangle is called a
centroid
5) An altitude of a triangle is a segment from a vertex to the line containing the
opposite side and perpendicular to the line containing that side.
a. Every triangle had three altitudes
b. The intersection point of the altitudes of a triangle is called the
orthocenter
5.2 Inequalities and Triangles
1) Exterior Angles Theorem: The measure of an exterior angle of a triangle is
equal to the sum of two remote interior angles
2) Exterior Angles Inequality Theorem: The measure of an exterior angles of a
triangle is greater than the measure of either of the two remote angles
3) Angle-Side Relationship: The longest angle of a triangle is equal to the longest
side of a triangle
5.3 Indirect Proofs
1) How to write Indirect proofs:
a. Assume that the conclusion is false
b. Show that the assumption leads to a contradiction
c. Because the assumption leads to a contradiction, that original
conclusion must be true.
5.4 The Triangle Inequality
1) Triangle Inequality Theorem: The sum of the lengths of any two sides of a
triangle is greater than the length of the third side.
5.5 Inequalities Involving Two Triangles
2 [Type text]
1) SAS (Side, Angles, Side) Inequality: Two sides of a triangle are congruent to
the two sides of another triangles. If the included angles in the first triangle
have a greater measure than the included angle in the second triangle, then
the third side od the first triangle is longer than the third side of the second
triangle.
2) SSS (Side, Side, Side) Inequality: If two triangle have two pairs of congruent
sides, then the angle opposite the longer 3rd side is the larger angles.
Chapter 6
6.1 Angles of Polygons
1) Concave and Convex Polygon:
a. Concave polygons- when the lines are extended, the sides goes into
the shape
b. Convex polygons: Is a polygon when the lines are extended, the
sides do not go into the center of the shape
2) Regular and Irregular Polygons:
a. Regular polygons: All the sides and all the angles of a polygon are
congruent
b. Irregular polygons: Not all sides and all the angles are congruent
3) Diagonal: A segment that connects two non-adjacent vertices of a polygon
4) Angle Sum Formula:
a. Interior Angles Sum Formula: 180(n-2) n= number of sides of a
regular polygon
b. Measure of each Interior Angle: 180(n-2)/2 n= number of angles of
a regular polygon
c. Measure of each exterior angle of regular polygons: m= 360/n
6.2 Parallelograms
1) Parallelogram: A quadrilateral with both pairs of opposite sides are
parallel
a. Opposite sides are parallel
b. Opposite sides are congruent
c. Opposite angles are congruent
d. Consecutive angles are supplementary
e. If it had one right angle, then it had four right angles
f. Diagonals bisect eachother
g. Each diagonal creates two congruent triangles
6.3 Tests for Parallelograms
1) 5 tests for parallelogram:
a. If both pairs of opposite sides are parallel, then it is a parallelogram
b. If both pairs of opposite sides are congruent, then it is a
parallelogram
c. If both pairs of opposite angles are congruent, then it is a
parallelogram
d. If diagonals bisect eachother, then it’s a parallelogram
e. If one pair of opposite sides are both congruent and parallel, then it
is a parallelogram
Clare Murphy-Putnam
6.4 Rectangles
1) Rectangle: A quadrilateral with four right angles
2) Properties of rectangles:
a. All properties of parallelograms
b. 4 right angles
c. Diagonals are congruent
3) Tests for Rectangles:
a. All tests for parallelograms
b. If both pairs of oppisite sides are parallel, then it is a rectangle
(use slope formula)
c. If diagonals bisect eachother, then it is a rectangle (use midpoint
and distance formula)
6.5 Rhombi & Squares
1) Rhombus: A quadrilateral with 4 congruent sides.
2) Properties of Rhombi:
a. All properties of Parallelograms
b. Diagonals are perpendicular
c. Each diagonal bisects a pair of opposite angles
3) Tests for Rhombi:
a. If both pairs of opposite sides are congruent, the it’s a rhombus (use
distance formula)
b. If diagonals bisect eachother, then it’s a rhombus (use midpoint and
slope formula)
4) Square: A quadrilateral that is both a rectangle and a rhombus
5) Properties of Square:
a. All properties of parallelograms
b. All properties of rectangles
c. All properties of rhombi
6) Tests for Squares:
a. If 4 sides and 4 right angles, then it’s a square (distance and slope
formulas)
6.6 Trapezoids
1) Trapezoid: Quadrelateral with one pair of parallel sides (bases)
1) 4 base angles
2) non-parallel sides are the legs
3) If the legs are congruent then it’s an isosceles trapezoid
i. Properties of Isosceles trapezoids:
ii. Each pair of base angles of an isosceles trapezoid are
congruent
iii. Diagonals of isosceles trapezoids are congruent
2) Segments that joins the midpoint of the legs is called the median
i. The median is parallel to the bases and its measure is half the
sum of the measure of the bases
3) Tests for Trapezoids:
1) If it has 1 pair of parallel sides, then it’s a trapezoid
4) Tests for Isosceles Trapezoids:
4 [Type text]
1) If the legs are congruent (use distance formula), upper or lower base
angles are congruent, and diagonals are congruent, then it’s an
isosceles triangle.
6.7 Coordinate Proofs & Quadrilaterals
1) Set up quadrilateral on coordinate plane with all coordinates names and
labels
2) Show formulas
3) Summarize steps
Isosceles Trapezoid
Square
Trapezoid
Parallelogram
Rectangle
Rhombus
Clare Murphy-Putnam
Ch5:
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1)
a. Given: D is on the bisector of angle CAE, DB is perpendicular to AC; DF is
perpendicular to EA
Prove: DB = DF
2)
In the figure, YW bisects angle XYZ, the measure of XYW = 6x + 2, and the measure of
angle ZYW = 8x – 6. Solve for x and find angle measure
3) Given: G is the centroid of triangle AEC, EB is perpendicular to AC, AB = 6, AF – 5,
AG = 5, and AE = CE CB = ?
6 [Type text]
4) What is the point of concurrency of the three lines containing the altitude of a
triangle called?
5) In triangle ABC, BD is congruent to DC. Point D lies on BC. What is AD?
Ch. 6
6) Given: Trapezoid WXYZ with midsegment ST. If WZ = 15 and ST = 20, find the
length of XY
7) Which quadrilateral is described by the following statement: It is equiangular
and equilateral
8) What is the missing reason in the following proof?
Statements
Reasons
1. BC is congruent to AD, AB is
1. Given
congruent to DC
2. ABCD is a parallelogram
2. If the opposite sides of a
quadrilateral are congruent, then it is
a parallelogram
3. BE = ED
3. _______________?________________
9) (1,1) and (6,4) are opposite vertices in a parallelogram. If (2,4) is the third
vertex, then the fourth vertex is _____________
10)Find the values of x, y, and z in parallelogram MNPQ
Clare Murphy-Putnam
Answers:
1) Angle Bisector Theorem – since D is on the bisector of angle CAE, and DB
is perpendicular to AC and DF is perpendicular to EA, DB is congruent to DF
though the angle bisector theorem
2) XYZ x=4, measure of angle XYZ = 52
6x + 2 = 8x – 6
8 = 2x x =4
6(4) + 2 = 26
8(4) - 6 = 26
26 + 26 = 52
3) 6- AB is congruent to CB because BE is a median
4) The orthocenter
5.) The median
6) 25-(15 + XY)/2 = 20
15 + XY = 40
XY = 25
7) Square (all same angles and all same sides)
8) Diagonals of a parallelogram bisect each other
9) (5,1)
10) x=110, y = 40, z = 30 opposite angles in parallelograms are congruent and
consecutive angles are supplementary